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Home , A Passage to Infinity, Bijaganita, History of trigonometry

e ISSN–0976–7959 Asian Volume 11 | Issue 1 | June, 2016 | 78-86 DOI : 10.15740/HAS/AS/11.1/78-86 Science Visit us | www.researchjournal.co.in

A CASE STUDY Mathematics in ancient India

NIRANJAN SWAROOP Department of Mathematics, Christ Church College, KANPUR (U.P.) INDIA (Email: [email protected])

ABSTRACT “India was the motherland of our race and the mother of Europe’s languages. India was the mother of our philosophy, of much of our mathematics, of the ideals embodied in Christianity... of self-government and democracy. In many ways, Mother India is the mother of us all.”- Will Durant (American Historian 1885-1981). Mathematics had and is playing a very significant role in development of Indian culture and tradition. Mathematics in Ancient India or what we call it Vedic Mathematics had begun early Iron Age. The SHATAPATA BRAHAMANA AND THE SULABHSUTRAS includes concepts of irrational numbers, prime numbers, rule of three and square roots and had also solved many complex problems. Ancient Hindus have made tremendous contributions to the world of civilization in various fields such as Mathematics, Medicine, Astronomy, , Botany, Metallurgy, Civil Engineering, and Science of consciousness which includes Psychology and philosophy of Yoga. The object of this paper tries is to make society aware about the glorious past of our ancient science which has been until now being neglected without any fault of it. Key Words : Ancient, Mathematics, Ganita, Sulabhsutras

View point paper : Swaroop, Niranjan (2016). Mathematics in ancient India. Asian Sci., 11 (1): 78-86, DOI : 10.15740/HAS/AS/11.1/78- 86.

ny account of the classical sciences of India has wrote:- to begin with mathematics, the reasons being, “Occidental mathematics has in past centuries Athe ancient Sanskrit text Vedanga Jyotisa (ca. broken away from the Greek view and followed a course fourth century B.C.E.) says, which seems to have originated in India” Like the crest on the peacock’s head, Like the gem The love affair between Indian culture and numbers in the cobra’s hood so stands mathematics at the head has been long. When most societies had difficulty in of all the sciences. handling numbers beyond 1000, the Buddhist text Lalita- The Sanskrit word used for mathematics in this vistara (before the fourth century C.E.) not only has no verse is ganita, which literally means “reckoning.” problem with huge numbers, but seems to revel in giving The unique Indian view about the classical of them names (10145, the highest number quoted, being mathematics is that number was treated as the primary called dhvaja-nis’a-mani.) concept. which we call today Ancient This thing has been substantiated by a distinguished Indian Mathematics has it’s origination in Indian Swiss mathematician-physicist in 1929 that when he subcontinent from 1200 BCE upto the end of the 18th

HIND INSTITUTE OF SCIENCE AND TECHNOLOGY NIRANJAN SWAROOP century. In this classical period of Indian mathematics Ancient Indian was aware of the concept of zero, (400 CE to 1600 CE), not only important but excellent actually they are solely responsible for hugely important contributions were made by scholars such as invention in mathematics and it would be no hyperbole , , Mahâvîra, Bhaskara II, to say that actually they invented the zero. Madhava of Sangamagrama and . The earliest recorded date an inscription of zero on The decimal number system which we use today Sankheda Copper Plate was found in Gujrat In 585- was first recorded in Indian mathematics. Indian 586CE , not only this but also use of a circle which mathematicians made early contributions to the study of symbolizes the number zero is found in a 9th Century the concept of zero as a number, negative numbers, engraving in a temple in Gwalior in central India. But arithmetic, and not only this but In addition, the brilliant conceptual leap to include zero as a number was further advanced in India and in in its own right (not merely as a placeholder, a blank or particular, the modern definitions of and cosine were empty space within a number, as it had been treated developed here also in India. until that time) is usually associated to the 7th Century These mathematical concepts were transmitted to Indian mathematicians Brahmagupta - or possibly the Middle East, , and Europe and led to further another Indian, Bhaskara I - even though it may well developments that now form the foundations of many have been in practical use for centuries before that. The areas of mathematics. use of zero as a number which could be used in Ancient and medieval Indian mathematical works, calculations and mathematical investigations would was composed in Sanskrit, which usually consisted of a revolutionize mathematics. section of sutras in which a set of rules or problems were Had there been no zero there would have been no stated with great beauty in verse in order to aid binary number system and as a result no computers and memorization by a student. This was followed by a how cumbersome would have been counting? second section consisting of a prose commentary The Hindu culture has positional number system in (sometimes multiple commentaries by different scholars) base ten they used a dot to represent an empty place, that explained the problem in more detail and provided sunya which meant empty was the name for this dot at justification for the solution. In the prose section, the this point earlier zero was a place holder and an aid in form (and therefore, its memorization) was not computation, by 500 C.E. the Hindus use a small circle considered as important as the ideas involved. to represent zero this circle was later on recognized as a All mathematical works were transmitted from one numeral zero. generation to other orally until approximately 500 BCE; thereafter, they were transmitted both orally and in The ancient Hindu symbol of a circle with a dot manuscript form. The oldest extant mathematical in the middle, known as bindu or bindhu, document produced on the Indian subcontinent is the symbolizing the void and the negation of the self, was probably instrumental in the use of a circle birch bark , discovered in 1881 in as a representation of the concept of zero. the village of Bakhshali, near Peshawar (modern day Pakistan) and is likely from the 7th century CE. Fig. 2 : Bindu

Zero in India The number 0 appear in the late 10th century in India. The concept of zero as a number and not merely a symbol for separation is attributed to India, where by the 9th century CE, practical calculation were carried out using zero, which was treated like any other number, even in case of division. The word for zero in Hindu-India is ‘Shunya” meaning void or empty. India is recognized with great respect for its invention of Zero by importance with technological world. These are the numbers that the Hindus and the Arabics use. Fig. 3 : Ancient Indian text explains concept of Infinity, as reciprocal of zero as well as they were also aware of Fig. 1 : The case of the “ theorem” the concept of

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As x gets bigger and bigger 1 approaches zero. Conversely, as x gets smaller and The approximation formula : x smaller and approaches zero, 1 approaches infinity x The formula is given in verses 17 – 19, Chapter VII, Mahabhaskariya of Bhaskara I. A translation of the 1 5 x verses is given below: 4 (Now) I briefly state the rule (for finding 3 the bhujaphala and the kotiphala, etc.) without making 2 use of the Rsine-differences 225, etc. Subtract the 1 degrees of a bhuja (or koti) from the degrees of a half 0 circle (that is, 180 degrees). Then multiply the remainder -1 by the degrees of the bhuja or koti and put down the -2 result at two places. At one place subtract the result -3 from 40500. By one-fourth of the remainder (thus, obtained), divide the result at the other place as multiplied -4 anthyaphala (that is, the epicyclic ). Thus, -5 by the ‘ is obtained the entirebahuphala (or, kotiphala) for the -5 -4 -4 -2 -1 0 1 2 3 4 5 , or the -. So also are obtained the direct and inverse Rsines. (The reference “Rsine-differences 225” is an allusion to Aryabhata’s sine table.) In modern mathematical notations, for an x in degrees, this formula gives : 4x (180 – x) sin x0  40500 – x (180 – x)

Equivalent forms of the formula : Bhaskara I’s sine approximation formula can be expressed using the radian measure of as follows (O’Connor and Robertson, 2000) : 16x( – x) sin x  2 5 – 4x( – x) For a positive integer n this takes the following form (George, 2009) : Fig. 4 : Sine approximation formula  16(n – 1) sin  n 5n2 – 4n  4 In mathematics, Bhaskara I’s sine approximation formula is a rational expression in one variable for Equivalent forms of Bhaskara I’s formula have been the computation of the approximate values of given by almost all subsequent astronomers and the trigonometric discovered byBhaskara I (c. 600 mathematicians of India. For example, Brahmagupta’s – c. 680), a seventh-century Indian mathematician. This (598 – 668 CE) Brhma-Sphuta-Siddhanta (verses 23 formula is given in his treatise titled Mahabhaskariya. – 24, Chapter XIV) (Gupta, 1967) gives the formula in It is not known how Bhaskara I arrived at his the following form: approximation formula. However, several historians of Rx (180 – x) R sin x0  mathematics have put forward different theories as to 1 10125 – x (180 – x) the method Bhaskara might have used to arrive at his 4 formula. The formula is elegant, simple and enables one Also, Bhaskara II (1114 – 1185 CE) has given this to compute reasonably accurate values of trigonometric formula in his Lilavati (Kshetra-vyavahara, Soka No.48) sines without using any geometry whatsoever. in the following form:

Asian Sci., 11(1) June, 2016 : 78-86 80 HIND INSTITUTE OF SCIENCE AND TECHNOLOGY NIRANJAN SWAROOP

4  2R  2Rx  (360R – 2Rx) 1 2R sin x0  1  5  (360 R)2 – 2Rx  (360R – 2Rx) 0.8 4 0.6

Accuracy of the formula : 0.4

1.2 0.2

0 1.0 0 20 40 60 80 100 120 140 160 180 x 0.8 x'(180 – x) x'(180 – x) 0.6 8100 9000 sin (x) 0.4 (x) in degress

0.2 Fig. 6 : Power

0 20 40 60 80 100 120 140 160 180 Table 1 : Trigonometric functions -0.2 x Function (abbreviation) Definition 4' x' (180 – x) 0.2 sin (x) + 0.2 – opposite a 40500 – x (180 – x) (x) in degress sine (sin) sin A  sin (x) hypotenuse c (x) in degress

adjacent b Fig. 5 : Figure illustrates the level of accuracy of the Bhaskara cosine (cos) cos A  I's sine approximation formula. The shifted curves 4 x hypotenuse c (180 - x) / (40500 - x (180 - x) - 0.2 and sin (x) + 0.2 look like exact copies of the curve sin ( x ). opposite a tangent (tan) tan A  adjacent b The formula is applicable for values of x° in the adjacent b range from 0 to 180. The formula is remarkably accurate cotangent (cot or ctn) cot A  in this range. The graphs of sin (x) and the approximation opposite a formula are indistinguishable and are nearly identical. hypotenuse c secant (sec) sec A  One of the accompanying figures gives the graph of the adjacent b error function, namely the function : cosecant (csc) hypotenuse c 4x (180 – x) cscA  sin x0  opposite a 40500 – x (180 – x) In using the formula. It shows that the maximum absolute error in using the formula is around 0.0016. From (Table 1). a plot of the percentage value of the absolute error, it is Golden Age Indian mathematicians made clear that the maximum percentage error is less than fundamental advances in the theory of trigonometry. 1.8. The approximation formula thus, gives sufficiently They used ideas like the sine, cosine and tangent accurate values of sines for all practical purposes. functions (which relate the angles of a to the However, it was not sufficient for the more accurate relative lengths of its sides) to survey the land around computational requirements of astronomy. The search them, navigate the seas and even chart the heavens. For for more accurate formulas by Indian astronomers instance, Indian astronomers used trigonometry to eventually led to the discovery the power series calculate the relative distances between the Earth and expansions of sin x and cos x by Madhava of the Moon and the Earth and the Sun. They realized that, Sangamagrama (c. 1350 – c. 1425), the founder of the when the Moon is half full and directly opposite the Sun, school of astronomy and mathematics. then the Sun, Moon and Earth form a right angled triangle, 1 Indian astronomers have used the concept of and were able to accurately measure the angle as /7°. trigonometry and its table to calculate the distance of sun Their sine tables gave a ratio for the sides of such a from earth they defined trigonometric functions as below triangle as 400:1, indicating that the Sun is 400 times

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Sun

10 7 1

400 Vertical gnomon

P  b Fig. 10 : Determination of the latitude of a place on equinox day Fig. 7 : Indian trigonometry tables showed that an angle of 1/ 70 indicate triangle sides with a ratio of 400:1, mean- ing that the sun is 400 times further away from the work. earth than the moon The accuracy with which Aryabhata calculated some astronomical results.

Ionosphere

BHASKARACHARYA II (1114-1183 CE) Genius in Algebra and Astronomy, Contributor to world math Skywave 600 km (e.g)

“In many ways, Bhasfkara represents the peak of mathematical and astronomical knowledge in the twelth centry. He reached an understanding of , astronomy, the number systems, and solving equations, which were not to be achieved anywhere ~ 9343 km else in the world for several centuries.” Rx Tx London Madagascar (e.g.) He wrote “Bijaganita”, a treatise on algebra, in which he derived a cyclic, ‘Cakraval’ method for solving equations of the form ax2 + bx + c = y, which is usually attributed Fig. 8 : Length of skywave propagation path is 9419 km Real to William Brouncker (1657). transmission latency is just 31.4 msec (9.4e6 / 300e6) His book “lilavati” covers many branches of yet a fake latency of ~250 msec is added to “satellite” mathematics, arithmetic, algebra, geometry, trigonometry and mensuration, and his comms. who decided that and why? treatise “Siddhant Shiromani” on astronomy contains several results in trigonometry and integral and differential calculus.

See http://www-groups.dcs.st-and.ac.uk/-history/Projects/Pearce/Chapters/Ch8 5.html   http://www.britannica.com/EBchecked/topic/64067/Bhaskara-II http://www.newworldencyclopedia.org/entry/Bh%C4%81skara II

Ujjain Fig. 11 : Bhaskaracharya II Sun vertical S R 1 Bhaskara II  Vertical Some of Bhaskara II's contributions are :  – A proof of the by X 1 calculating the same area in two different ways and then canceling out terms to get a2 + b2 = c2. Lanka – Solution of quadratic, cubic and quartic indeterminate equation are explained. – Solutions of indeterminate quadratic equations 2 2 Center of land and water (of the type ax + b = y ). – Integer solutions of linear and quadratic Fig. 9 : Angles made at noon at and X1 indeterminate equations. – Solved quadratic equations with more than one unknown, and found negative and irrational further away from the Earth than the Moon. solutions. – In Siddhanta Shiromani, Bhaskara developed along with a number of Some mathematician of ancient India : other trigonometric results. Some mathematicians of ancient India and their Fig. 12 : Bhaskara II

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Brahmagupta gave the first rules for dealing with zero as a number Bhaskara I (600-680 AD) When zero is added to a number or subtracted from a number, the number remains unchanged. A number multiplied by zero becomes zero.

... and for dealing with positive (fortune) and negative (debt) numbers y A dept minus zero is a debt. A fortune minus zero is a fortune. 1 sin  1 sin   x  Zero minus zero is a zero. cos  cos  A debt subtracted from zero is a fortune. A fortune subtracted from zero is a debt. The product of zero multiplied by a debt or fortune is zero. The product of zero multiplied by zero is zero. The product or quotient of two fortunes is one fortune. The product or quotient of two debts is one fortune. 16x(π  x) π The product or quotient of a debt and a fortune is a debt. sinx  (0  x  ) The product or quotient of a fortune and a debt is a debt. 5π2  4x(π  x)' 2 Fig. 13 : First rules for dealing with zero as a number Fig. 16 : Bhaskara I (600-680 AD)

meaning ‘treatise from the Ashmaka’. This treatise is also referred to as ‘Ayra-shatas-ashta’ which translates to ‘Aryabhata’s 108’. This is a very literal name because the treatise did in fact consist of 108 verses. It covers several branches of mathematics such as algebra, arithmetic, plane and spherical trigonometry. Also included in it are theories on continued fractions, sum of power series, sine tables and quadratic equations. Aryabhata worked on the place value system using letters to signify numbers and stating qualities. He also came up with an approximation of ( ) and area of a triangle. He introduced the concept of sine in his work Fig. 14 : Aryabhata (499 A.D.) called ‘Ardha-jya’ which is translated as ‘half-’. One of the most significant input of Brahmagupta Aryabhata wrote many mathematical and to mathematics was the introduction of ‘zero’ to the astronomical treatises. His chief work was the number system which stood for ‘nothing’. His work the ‘Brahmasphutasiddhanta’ contained many mathematical Astronomy : Aryabhata findings written in verse form. It had many rules of

Fig. 15 : Astronomy : Aryabhata

‘Ayrabhatiya’ which was a compilation of mathematics and astronomy. The name of this treatise was not given to it by Aryabhata but by later commentators. A disciple by him called the ‘Bhaskara’ names it ‘Ashmakatanra’ Fig. 16 : Brahmagupta

Asian Sci., 11(1) June, 2016 : 78-86 83 HIND INSTITUTE OF SCIENCE AND TECHNOLOGY MATHEMATICS IN ANCIENT INDIA arithmetic which is part of the mathematical solutions our great heritage was only basis for their recognition now. These are ‘A positive number multiplied by a and western historians as well, although the situation is positive number is positive.’, ‘A positive number multiplied improving a little bit now. Two questions, firstly, why have by a negative number is negative’, ‘A negative number Indian works been neglected and the secondary question, multiplied by a positive number is negative’ and ‘A shouldn’t this neglect be considered a great injustice? negative number multiplied by a negative number is These has to be answered then and only then we will be positive’. The book also consisted of many geometrical able appreciate and revive what had been done by our theories like the ‘Pythagorean Theorem’ for a right angle mathematicians of Ancient India. triangle. Brahmagupta was the one to give the area of a What I primarily wished to tackle was to answer triangle and the important rules of trigonometry such as that is, what appears to have been the motivations and values of the sin function. He introduced the formula for aims of scholars who have contributed to the Eurocentric cyclic . He also gave the value of ‘Pi’ as view of mathematical history. This leads to the way square root ten to be accurate and 3 as the practical ancient Indian Mathematics has been neglected and value. Additionally he introduced the concept of negative ignored by Eurocentric scholars and scientist who numbers. thought Indians pagans religious and Mathematics which Brahmagupta solved the so-called Pell equation nx2 is mentioned in their texts etc could not be the basis of + 1 = y2. their monotheistic culture, and these scholars believed In modern notation, the solution he gave was that this is true ever since, but the modern researches 2t t2  n and evidences are proving that vast majority of works x  and y  t2 – n t2 – n has been done by Indians in ancient age and most of the where, t could be replaced by any number. books are written in Sanskrit or Hindi. For example, if t = 3, the we get the solution In last I only want to say that there is a vast scope 6 9 n for study of VEDIC MATHEMATICS and it should be x  and y  9– n 9 – n , so that : definitely studied only because there are much evidences

2 that Indian mathematics has played a very important role  6  36n nx2  1  n   1  in the , much more work should   2  9 – n  81 – 18n  n be done to analyze ancient documents and archeological 2 2 2 36n  81 – 18n  n 81  18n  n  9  n  2  1       y evidences of a very intelligent society. 81 18n  n2 81 – 18n  n2  9 – n  The , a textbook on astronomy of REFERENCES ancient India which was last complied in 1000B.C. and Al-Daffa, A.A. (1977). The muslim contribution to is believed to be handed down from 3000 B.C. by aid of mathematics. [All references denoted by AA’D] Humanities complex mnemonic recital method still known today , it Press, U.S.A. clearly shows the Earth’s to be 7,840 miles as Almeida, D.F., John, J.K. and Zadorozhnyy, A. (2001). Keralese compared to modern method which found it be 7926,7 Mathematics: Its possible transmission to Europe and the miles , it also shows distance between Earth and Moon consequential educational implications. J. Nat. Geometry, 20 to be 253,000 miles whereas the same distance as : 77-104. calculated by modern methods comes out too be 252,710 Boyer, C.B. (1968). A history of mathematics. John Wiley and miles . Sons, INC, U.S.A. It clearly shows the high accuracy acquired by the mathematician of ancient India Channabasappa, M.N. (1984). Mathematical terminology peculiar to the Bakhshali manuscript. Ganita- Bharati (Bulletin of the Indian Society of the History of Conclusion : Mathematics) 6 : 13-18 I wish to conclude firstly by simply saying that the work of Indian mathematicians has been severely Datta, B. and Singh, A.N. (1962). History of hindu neglected by our own historians (so called secular and mathematics, a source book, Parts 1 and 2, (single volume). [Denoted by AS/BD] Asia Publishing House, Bombay (M.S.) left oriented) and in whose view neglecting and abusing INDIA.

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WEBLIOGRAPHY MacTutor History of Mathematics: History Topics and Mathematical biographies Fertilizer Industry, India Fertilizer Industry (2010, June, 29). (Written and compiled by Professor Edmund F Robertson Kalyanaraman, S. The Sarasvati-Sindhu Civilization (c. 3000 and Dr John J O’Connor, University of St Andrews. I used 38 B.C.). Article retrieved from Indology List-Serve. articles form this website.).

Received : 16.04.2016; Accepted : 28.05.2016

Asian Sci., 11(1) June, 2016 : 78-86 86 HIND INSTITUTE OF SCIENCE AND TECHNOLOGY